1. Field of the Invention
The present invention relates to a signal processing method for eliminating the aliasing occurring due to sampling so as to broaden the band of a signal, more particularly relates to a signal processing method suitable for use when performing a plurality of sampling operations having different phase differences on a pseudo-periodic signal (pseudo static image) having a high correlation occurring at a constant period, such as a television signal, or an identical signal so as to obtain a higher resolution signal (static image).
2. Description of Related Art
In general, when sampling a continuous analog signal on a time basis by a sampling frequency fs to produce a digital signal, the sampling theorum results in the frequency component of the continuous signal above the Nyquist frequency (fs/2) being transformed to the low band side and causes so-called aliasing distortion. Therefore, to avoid this distortion, normally low band pass filtering is performed before the sampling so as to remove the frequency component above fs/2.
The characteristics of this low band pass filtering, however, are not at all ideal. Some frequency component above fs/2 even ends up remaining. As a result, aliasing occurs due to the sampling. This aliasing, for example, in the case of an image, causes the phenomenon of what originally should be a slanted line appearing as steps.
Further, for example, a large number of pixels CCD is expensive, so when using a small number of pixels CCD, the sampling interval becomes larger and the sampling frequency becomes lower. In this case as well, the aliasing is suppressed by cutting the frequency component above fs/2, so only the extremely low frequency component remains and only a blurry image is obtained. That is, in this case, a slanted line does not become steps, but sharp lines end up become blurry edged belts.
The following techniques have been proposed to obtain a high resolution signal with a high sampling frequency from a signal with a low sampling frequency.
This technique simply raises the sampling frequency. That is, it creates a new sampling point between sampling points of the original signal. Among the simpler forms, the technique called 0-th hold which repeats the previous value as it is, the linear interpolation method using the linearly interpolated value of the previous and subsequent values, and the techniques called B-spline or cubic convolution are well known. These techniques enable the number of the sampling points to be increased, so for example are suitable for enlarging the display of an image etc.
This technique reproduces the high band not by simple interpolation, but by estimating what the original analog signal including the high frequency component was like at the same time as raising the sampling frequency.
If an image, for example there is the technique of performing linear interpolation, then changing the pixel values based on the hypothesis that the pixel values of a natural image inherently change smoothly and repeating this several times to reconstruct the high frequency component.
Further, there is the technique of outputting a high-definition signal when an NTSC signal is input by learning in advance the correspondence between the pixels in high definition signals and corresponding pixels in the NTSC signal or several nearby pixels.
Further, there is the technique of outputting the corresponding broad band signal when a narrow band signal is input by learning in advance in the same way for an audio signal.
FIGS. 16A to 16D are views for explaining the sub-Nyquist sampling method.
FIG. 16A shows the spectrum of a sampled signal. The ordinate shows the signal level, while the abscissa shows the frequency of the signal.
In general, if a sampled signal having the spectrum shown in FIG. 16A is sampled at a sampling frequency fs, a signal having the spectrum shown in FIG. 16B is obtained. In the figure, the hatched portion shows aliasing where the component of the sampled signal above the frequency fs/2 is transformed into a low band.
To suppress this aliasing, first, two series of sampling having a 180 degree relative phase difference are performed on the sampled signal having the spectrum shown in FIG. 16A.
As a result, signals having the spectra shown in FIGS. 16C and 16D are obtained.
Here, the odd number order modulation component of the spectrum of the signal shown in FIG. 16D becomes opposite in phase from the odd number order modulation component of the spectrum of the signal shown in FIG. 16C.
Therefore, by adding the signal having the spectrum shown in FIG. 16C and the signal having the spectrum shown in FIG. 16D, a signal having the spectrum shown in FIG. 16E is obtained.
That is, it is possible to cancel out the primary modulation component included in the signal having the spectrum shown in FIG. 16C and the primary modulation component included in the signal having the spectrum shown in FIG. 16D and thereby eliminate the aliasing. As a result, the signal band can be doubled.
In this sub-Nyquist sampling method, when the phase difference between the two series of sampling operations is 180 degrees, the aliasing can be removed and the signal band doubled. Further, even when performing n number of series of sampling operations with phase differences between adjoining operations of 360/n degrees, it is similarly possible to eliminate the aliasing occurring at the sampling operations and increase the signal band n-fold.
In the above sub-Nyquist sampling method, the aliasing was eliminated conditional on the phase difference of the two series of sampling operations being 180 degrees. In this technique, however, aliasing can be eliminated even if the phase difference of the sampling operations is other than 180 degrees.
In this technique, note is taken of the fact that the signal observed as aliasing is due to the M number of imaging (high harmonic) components occurring due to the sampling. The H number of imagings are removed by establishing simultaneous equations by preparing (M+1) number of images.
For example, by sampling a signal having a frequency component up to two times the Nyquist frequency and preparing three digital images including aliasing, a component up to two times the original Nyquist frequency is reconstructed.
The imaging component ends up overlapping the basic spectrum since the sampling frequency is low. This cannot be broken down, so the result is aliasing. Therefore, if the basic spectrum is found, the aliasing is removed.
Specifically, first, the digital signals E0, E1, and E2 of the three images, as shown by FIGS. 17A to 17C, are combinations of the basic frequency F(xcfx89), the primary imaging F(xcfx89xe2x88x92xcfx89S), and the negative primary imaging F(xcfx89+xcfx89S)
Due to the sampling phase difference, each term is subject to an exp term. This can be expressed by the following equations (1).
E0=F(xcfx89+xcfx89S)+F(xcfx89)+F(xcfx89xe2x88x92xcfx89S)E1=
xe2x80x83exp(jxcex11)F(xcfx89+xcfx89S)+F(xcfx89)+exp(xe2x88x92jxcex11)F
(xcfx89xe2x88x92xcfx89S)E2=
exp(Jxcex12)F(xcfx89+xcfx89S)+F(xcfx89)+exp(xe2x88x92jxcex12)F
(xcfx89xe2x88x92xcfx89S) . . . xe2x80x83xe2x80x83(1)
In the above equations (1), xcex11 and xcex12 are phase differences (rad) with the digital signal E0. These are detected by a known detection method and are considered known. Further, xcfx89S shows the sampling frequency.
If it were possible to use the above equations (1) to eliminate the imaging (xcfx89xe2x88x92xcfx89S) and F(xcfx89+xcfx89S) and leave only the frequency F(xcfx89), then the aliasing could be cancelled.
Here, if the weighting coefficients w0, w1, and w2 are multiplied with the equations (1), the following equations (2) are obtained.
w0E0=w0F(xcfx89+xcfx89S)+w0F(xcfx89)+w0F
(xcfx89xe2x88x92xcfx89S)w1E1=w1exp(jxcex11)F(xcfx89+xcfx89S)+w1F
(xcfx89)+w1exp(xe2x88x92jxcex11)F(xcfx89+xcfx89S)w2
E2=w2exp(jxcex12)F(xcfx89+xcfx89S)+w2F
(xcfx89)+w2exp(xe2x88x92jxcex12)F(xcfx89xe2x88x92xcfx89S)xe2x80x83xe2x80x83(2)
where,                                                         ∑              i                        ⁢                          xe2x80x83                        ⁢                          w              i                                =          1                ⁢                  xe2x80x83                ⁢                  
                ⁢                                            ∑              i                        ⁢                          xe2x80x83                        ⁢                                          w                i                            ⁢              exp              ⁢                              xe2x80x83                            ⁢                              (                                  jα                  i                                )                                              =          0                ⁢                  xe2x80x83                ⁢                  
                ⁢                              that            ⁢                          xe2x80x83                        ⁢            is                    ,                      
                    ⁢                                                    ∑                i                            ⁢                              xe2x80x83                            ⁢                                                w                  i                                ⁢                                  xe2x80x83                                ⁢                cos                ⁢                                  xe2x80x83                                ⁢                                  (                                      α                    i                                    )                                                      =            0                          ⁢                  xe2x80x83                ⁢                  
                ⁢                  xe2x80x83                ⁢        and        ⁢                  xe2x80x83                ⁢                  
                ⁢                                            ∑              i                        ⁢                          xe2x80x83                        ⁢                                          w                i                            ⁢                              xe2x80x83                            ⁢              sin              ⁢                              xe2x80x83                            ⁢                              (                                  α                  i                                )                                              =          0                                    (        3        )            
Here, by determining the weighting coefficients w0, w1, and w2 so as to satisfy the above equations (2), it is possible to solve the simultaneous equations to cancel all of the imaging terms and leave only the term of the basic frequency F(xcfx89) and possible to find the basic frequency F(xcfx89) shown in FIG. 17D from equation (4). Due to this, the aliasing can be removed and the band broadened. Here, WL is a real number.
F(xcfx89)=w0E0+w1E1+w2E2 . . . xe2x80x83xe2x80x83(4)
This technique is disclosed in Japanese Unexamined Patent Publication (Kokai) No. 7-245592 of the same assignee and cancels the aliasing from the two input signals (input image) including frequency components up to two times the Nyquist frequency to reconstructs the frequency component up to two times the Nyquist frequency.
This technique also cancels the imaging component in the same way as the technique disclosed in the above Japanese Unexamined Patent Publication (Kokai) No. 8-336046. It not only adjusts the gain of the input image, but also shifts the phase.
Here, the xe2x80x9cphase shiftxe2x80x9d is processing for changing the phase characteristics while passing the entire band as in for example Hilbert transformation. When viewing the spectrum on a complex plane, gain adjustment is an operation for changing the absolute value of a vector, while phase shift is an operation for rotating the vector.
As clear from the law of addition of vectors, to cancel a vector exhibiting imaging by just gain adjustment, when the angle formed by the vectors is other than 180 degrees, two vectors are not enough. A minimum of three vectors becomes necessary.
As opposed to this, if performing a phase shift in addition to gain adjustment, as shown in FIGS. 18A to 18C, no matter what the angle is between the vectors of the digital signals E0 and E1, it is possible to make the angle 180 degrees by rotation and possible to cancel the imaging vector by two vectors. That is, if an image signal, two input images are enough.
Summarizing the problems to be solved the invention, in the above interpolation method, there is the problem that the high frequency component of above fs/2 lost once in sampling from a continuous signal cannot be reconstructed and only a blurry image or distorted image can be obtained.
Further, in the above prediction technique, there is the problem that the probability of accurate prediction is still low.
Further, in the above sub-Nyquist sampling method, there is the problem that it is only possible to eliminate the aliasing when the phase difference in the two series of sampling operations is 180 degrees.
Further, in the technique disclosed in the above Japanese Unexamined Patent Publication (Kokai) No. 8-33604, at least three images were necessary in the case of a one-dimensional signal, so reproduction was not possible when that number of images could not be prepared. Further, in the technique disclosed in Japanese Unexamined Patent Publication (Kokai) No. 9-69755 applying this technique to a two-dimensional signal, nine signals are required in order to reproduce up to double the Nyquist frequency in two dimensions for the vertical and horizontal directions. Therefore, for example, if trying to remove the aliasing due to high order imaging components, the number of necessary images becomes extremely large. It becomes difficult to prepare that many images.
Further, in the phase shift technique disclosed in the above Japanese Unexamined Patent Publication (Kokai) No. 7-245592, it is described to reduce the number of images necessary, but it is not disclosed specifically how to remove the high order imaging components. Further, the above publication does not disclose the case where the phase difference of the sampling operations is two-dimensional.
An object of the present invention is to provide a signal processing method suitably removing even the higher order imaging components causing aliasing included in a digital signal obtained by sampling so as to reproduce a broad band signal free of aliasing.
According to a first aspect of the present invention, there is provided a signal processing method for removing from a plurality of discrete signals, obtained by sampling an identical or substantially identical continuous signal using sampling phases differing in one-dimensional or two-dimensional directions and including a basic spectral component and imaging components other than the basic spectral component included in the continuous signal, the imaging components and generating a signal in accordance with the basic spectral component, comprising the steps of inputting the plurality of discrete signals including aliasing and differing in sampling phases and the value of the sampling phase difference, generating complex simultaneous equations having the basic spectral component and imaging components as variables, having an amount determined by the phase difference as a coefficient of the imaging components, and having the frequency domain expression of the input signal as a constant, solving the complex simultaneous equations, transforming the input signals to the frequency domain, multiplying complex numbers with the frequency domain expressions of the input signals based on the solutions to the complex simultaneous equations, and adding the results of the multiplication of the complex numbers with the frequency domain expressions of the input signals so as to obtain a signal free of aliasing.
Preferably, the sampling phase difference is one-dimensional.
Alternatively, preferably the method where the signals and the sampling phase difference are two-dimensional, further includes the steps of dividing them into a region where both x and y are positive frequencies and a region where x and y are opposite in sign, generating simultaneous equations for these divided regions, solving the generated simultaneous equations, multiplying complex numbers with the frequency domain expressions of the input signals based on the solutions of the simultaneous equations, and adding the results of multiplication of the complex numbers with the frequency domain expressions of the input signals, and adding the results of addition of the regions where both x and y are positive frequencies and the results of addition of the regions where x and y are opposite in sign to obtain a signal free of aliasing.
According to a second aspect of the present invention, there is provided a signal processing method for removing from a plurality of discrete signals, obtained by sampling an identical or substantially identical continuous signal using sampling phases differing in one-dimensional or two-dimensional directions and including a basic spectral component and imaging components other than the basic spectral component included in the continuous signal, the imaging components and generating a signal in accordance with the basic spectral component, including the steps of inputting the plurality of discrete signals including aliasing and differing in sampling phases and the value of the sampling phase difference, generating complex simultaneous equations having the basic spectral component and imaging components as variables, having an amount determined by the phase difference as a coefficient of the imaging components, and having the frequency domain expression of the input signal as a constant, solving the complex simultaneous equations, generating signals obtained by transforming the phases of the input signals, finding the linear sum of the input signals and signals obtained by transforming the phases of the input signals based on the solutions to the complex simultaneous equations, and adding the results of the linear sums of the input signals to obtain a signal free of aliasing.
Preferably, the sampling phase difference is one-dimensional.
Alternatively, preferably the method where the signals and the sampling phase difference are two-dimensional, further includes the steps of dividing them into a region where both x and y are positive frequencies and a region where x and y are opposite in sign, generating signals obtained by transforming the phases of the signals of the divided regions of the input signals, generating simultaneous equations for the divided regions, solving the generated simultaneous equations, and finding the linear sum of the input signals and the signals obtained by transforming the phases of the input signals based on the solutions of the simultaneous equations, and adding the results of the linear sum of the regions where both x and y are positive frequencies and the results of the linear sum of the regions where x and y are opposite in sign to obtain a signal free of aliasing.